Numerical study of bell–shaped solitons solutions for a generalized modified CH–DP equation

XiuMin Lyu; Qian Ge; XiuMin Lyu; Qian Ge

doi:10.3934/era.2025207

Electronic Research Archive

2025, Volume 33, Issue 8: 4603-4624. doi: 10.3934/era.2025207

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Numerical study of bell–shaped solitons solutions for a generalized modified CH–DP equation

XiuMin Lyu ¹,
Qian Ge ^{2
,
,}

1.
School of Science, Shandong Jiaotong University, Jinan 250357, China
2.
School of Science, Shandong Jianzhu University, Jinan 250101, China

Received: 27 March 2025 Revised: 28 June 2025 Accepted: 22 July 2025 Published: 07 August 2025

In this paper, we developed a numerical scheme based on barycentric rational interpolation to solve the generalized modified Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Using the proposed approach, we applied a direct linearization technique to transform the nonlinear generalized modified CH–DP equation into an equivalent linear formulation. Approximate solutions were constructed using barycentric rational interpolation basis functions. The spatiotemporal domain was discretized via barycentric rational interpolation, and a corresponding differentiation matrix was derived. Furthermore, theoretical analysis of error distribution and convergence properties was presented. Analytical results and numerical experiments validate the scheme's computational efficiency and high accuracy in solving the generalized modified CH–DP equation.
- numerical study,
- error estimate,
- direct linearization,
- barycentric rational collocation method,
- generalized modified CH–DP equation
Citation: XiuMin Lyu, Qian Ge. Numerical study of bell–shaped solitons solutions for a generalized modified CH–DP equation[J]. Electronic Research Archive, 2025, 33(8): 4603-4624. doi: 10.3934/era.2025207

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Abstract

In this paper, we developed a numerical scheme based on barycentric rational interpolation to solve the generalized modified Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Using the proposed approach, we applied a direct linearization technique to transform the nonlinear generalized modified CH–DP equation into an equivalent linear formulation. Approximate solutions were constructed using barycentric rational interpolation basis functions. The spatiotemporal domain was discretized via barycentric rational interpolation, and a corresponding differentiation matrix was derived. Furthermore, theoretical analysis of error distribution and convergence properties was presented. Analytical results and numerical experiments validate the scheme's computational efficiency and high accuracy in solving the generalized modified CH–DP equation.

References

[1]	M. A. Yousif, B. A. Mahmood, F. H. Easif, A new analytical study of modified Camassa-Holm and Degasperis-Procesi equations, Am. J. Comput. Math., 5 (2015), 267–273. https://doi.org/10.4236/AJCM.2015.53024 doi: 10.4236/AJCM.2015.53024
[2]	A. Golbabai, M. Javidic, A spectral domain decomposition approach for the generalized Burgers-Fisher equation, Chaos, Solitons Fractals, 39 (2009), 385–392. https://doi.org/10.1016/j.chaos.2007.04.013 doi: 10.1016/j.chaos.2007.04.013
[3]	S. H. Hashemi, H. R. Mohammadi-Daniali, D. D. Ganji, Numerical simulation of the generalized Huxley equation by he's homotopy perturbation method, Appl. Math. Comput., 192 (2007), 157–161. https://doi.org/10.1016/j.amc.2007.02.128 doi: 10.1016/j.amc.2007.02.128
[4]	C. G. Zhu, R. H. Wang, Numerical solution of Burgers' equation by cubic B-spline quasi-interpolation, Appl. Math. Comput., 208 (2009), 260–272. https://doi.org/10.1016/j.amc.2008.11.045 doi: 10.1016/j.amc.2008.11.045
[5]	V. Gupta, M. K. Kadalbajoo, A singular perturbation approach to solve Burgers-Huxley equation via monotone finite difference scheme on layer-adaptive mesh, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1825–1844. https://doi.org/10.1016/j.cnsns.2010.07.020 doi: 10.1016/j.cnsns.2010.07.020
[6]	M. Sari, G. Gürarslan, I. Dag, A compact finite difference method for the solution of the generalized Burgers-Fisher equation, Numer. Methods Partial Differ. Equations, 26 (2010), 125–134. https://doi.org/10.1002/num.20421 doi: 10.1002/num.20421
[7]	Y. Huang, F. Mohammadi-Zadeh, M. H. Noori-Skandari, H. Ahsani-Tehrani, E. Tohidi, Space-time Chebyshev spectral collocation method for nonlinear time-fractional Burgers equations based on efficient basis functions, Math. Methods Appl. Sci., 44 (2021), 4117–4136. https://doi.org/10.1002/mma.7015 doi: 10.1002/mma.7015
[8]	Y. Yang, J. D. Wang, S. Y. Zhang, E. Tohidi, Convergence analysis of space-time Jacobi spectral collocation method for solving time-fractional Schrödinger equations, Appl. Math. Comput., 387 (2020), 124489. https://doi.org/10.1016/j.amc.2019.06.003 doi: 10.1016/j.amc.2019.06.003
[9]	F. Mohammadizadeh, S. G. Georgiev, G. Rozza, E. Tohidi, S. Shateyi, Numerical solution of $\psi$-Hilfer fractional black-scholes via space-time spectral collocation method, Alexandria Eng. J., 71 (2023), 131–145. https://doi.org/10.1016/j.aej.2023.03.007 doi: 10.1016/j.aej.2023.03.007
[10]	Z. Chen, H. Zhang, H. Chen, ADI compact difference scheme for the two-dimensional integro-differential equation with two fractional Riemann-Liouville integral kernels, Fractal Fract., 8 (2024), 707. https://doi.org/10.3390/fractalfract8120707 doi: 10.3390/fractalfract8120707
[11]	J. Lenells, Traveling wave solutions of the Camassa-Holm equation, Int. J. Differ. Equations, 217 (2005), 393–430. https://doi.org/10.1016/j.jde.2004.09.007 doi: 10.1016/j.jde.2004.09.007
[12]	A. Parker, Y. Matsuno, The peakon limits of soliton solutions of the Camassa-Holm equation, J. Phys. Soc. Jpn., 75 (2006), 124001. https://doi.org/10.1143/JPSJ.75.124001 doi: 10.1143/JPSJ.75.124001
[13]	G. P. Zhang, M. Zhang, New explicit exact traveling wave solutions of Camassa-Holm equation, Appl. Anal., 104 (2021), 69–81. https://doi.org/10.1080/00036811.2021.1986024 doi: 10.1080/00036811.2021.1986024
[14]	H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169–198. https://doi.org/10.1007/s00332-006-0803-3 doi: 10.1007/s00332-006-0803-3
[15]	A. Constantin, R. I. Ivannov, J. Lenells, Inverse svattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559–2575. https://doi.org/10.1088/0951-7715/23/10/012 doi: 10.1088/0951-7715/23/10/012
[16]	B. F. Feng, K. Maruno, Y. Ohta, On the $\tau$-functions of the Degaspeirs-Procesi equation, J. Phys. A Math. Theor., 46 (2013), 045205. https://doi.org/10.1088/1751-8113/46/4/045205 doi: 10.1088/1751-8113/46/4/045205
[17]	M. Alquran, M. Ali, I. Jaradat, N. Al-Ali, Changes in the physical structures for new versions of the Degasperis-Procesi-Camassa-Holm model, Chin. J. Phys., 71 (2021), 85–94. https://doi.org/10.1016/j.cjph.2020.11.010 doi: 10.1016/j.cjph.2020.11.010
[18]	A. H. Abdel-Kader, M. S. Abdel-Latif, New soliton solutions of the CH–DP equation using Lie symmetry method, Mod. Phys. Lett. B, 32 (2018), 1850234. https://doi.org/10.1142/S0217984918502342 doi: 10.1142/S0217984918502342
[19]	S. Lafortune, D. E. Pelinovsky, Spectral instability of peakons in the b-family of the Camassa-Holm equations, SIAM J. Math. Anal., 54 (2021), 4572–4590. https://doi.org/10.1137/21M1458776 doi: 10.1137/21M1458776
[20]	X. B. Sun, J. B. Li, G. R. Chen, Bifurcations, exact peakon, periodic peakons and solitary wave solutions of generalized Camassa-Holm-Degasperis-Procosi type equation, Int. J. Bifurcation Chaos, 33 (2023), 2350124. https://doi.org/10.1142/S0218127423501249 doi: 10.1142/S0218127423501249
[21]	M. Zhu, L. J. Guo, Y. Wang, Existence and uniqueness of the global conservative weak solutions to the cubic Camassa-Holm-Type equation, Commun. Nonlinear Sci., 132 (2024), 107903. https://doi.org/10.1016/j.cnsns.2024.107903 doi: 10.1016/j.cnsns.2024.107903
[22]	K. Grunert, Uniqueness of dissipative solutions for the Camassa-Holm equation, J. Differ. Equations, 412 (2024), 474–528. https://doi.org/10.1016/j.jde.2024.08.036 doi: 10.1016/j.jde.2024.08.036
[23]	Z. R. Liu, Z. Y. Ouyang, A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations, Phys. Lett. A, 366 (2007), 377–381. https://doi.org/10.1016/j.physleta.2007.01.074 doi: 10.1016/j.physleta.2007.01.074
[24]	R. Behera, M. Mehra, Approximate solution of modified Camassa-Holm and Degasperis-Procesi equations using wavelet optimized finite difference method, Int. J. Wavelets Multiresolution Inf. Process., 11 (2013), 1350019. https://doi.org/10.1142/S0219691313500197 doi: 10.1142/S0219691313500197
[25]	A. Kaur, V. Kanwar, H. Ramos, A coupled scheme based on uniform algebraic trigonometric tension B-spline and a hybrid block method for Camassa-Holm and Degasperis-Procesi equations, Comput. Appl. Math., 43 (2024), 16. https://doi.org/10.1007/s40314-023-02530-4 doi: 10.1007/s40314-023-02530-4
[26]	C. R. Qi, L. P. Xue, Q. F. Zhang, Error estimate of an invariant-preserving fourth-order scheme for the Degasperis-Procesi equation with smooth solutions, Appl. Math. Lett., 171 (2025), 109638. https://doi.org/10.1016/j.aml.2025.109638 doi: 10.1016/j.aml.2025.109638
[27]	M. S. Floater, K. Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math., 107 (2007), 315–331. https://doi.org/10.1007/s00211-007-0093-y doi: 10.1007/s00211-007-0093-y
[28]	J. P. Berrut, G. Klein, Recent advances in linear barycentric rational interpolation, J. Comput. Appl. Math., 259 (2014), 95–107. https://doi.org/10.1016/j.cam.2013.03.044 doi: 10.1016/j.cam.2013.03.044
[29]	A. P. Austin, K. Xu, On the numerical stability of the second barycentric formula for trigonometric interpolation in shifted equispaced points, IMA J. Numer. Anal., 37 (2017), 1355–1374. https://doi.org/10.1093/imanum/drw038 doi: 10.1093/imanum/drw038
[30]	A. Gillette, A. Rand, C. Bajaj, Error estimates for generalized barycentric interpolation, Adv. Comput. Math., 37 (2012), 417–439. https://doi.org/10.1007/s10444-011-9218-z doi: 10.1007/s10444-011-9218-z
[31]	S. Torkaman, M. Heydari, G. B. Loghmani, Piecewise barycentric interpolating functions for the numerical solution of Volterra integro-differential equations, Math. Methods Appl. Sci., 45 (2022), 6030–6061. https://doi.org/10.1002/mma.8154 doi: 10.1002/mma.8154
[32]	H. Y. Liu, J. Huang, Y. B. Pan, J. P. Zhang, Barycentric interpolation collocation method for solving linear and nonlinear high-dimensional Fredholm integral equations, J. Comput. Appl. Math., 327 (2018), 141–154. https://doi.org/10.1016/j.cam.2017.06.004 doi: 10.1016/j.cam.2017.06.004
[33]	J. Li, Y. L. Cheng, Linear barycentric rational collocation method for solving heat conduction equation, Numer. Methods Partial Differ. Equations, 37 (2021), 533–545. https://doi.org/10.1002/num.22539 doi: 10.1002/num.22539
[34]	J. Li, Linear barycentric rational collocation method for solving biharmonic equation, Demonstr. Math., 55 (2022), 587–603. https://doi.org/10.1515/dema-2022-0151 doi: 10.1515/dema-2022-0151
[35]	J. Li, Barycentric rational collocation method for fractional reaction-diffusion equation, AIMS Math., 8 (2023), 9009–9026. https://doi.org/10.3934/math.2023451 doi: 10.3934/math.2023451
[36]	J. Li, Y. L. Cheng, Barycentric rational interpolation method for solving 3 dimensional convection-diffusion equation, J. Approximation Theory, 304 (2024), 106106. https://doi.org/10.1016/j.jat.2024.106106 doi: 10.1016/j.jat.2024.106106
[37]	S. P. Li, Z. Q. Wang, Meshless Barycentric Interpolation Collocation Method-Algorithmics, Programs & Applications in Engineering, Science Press, 2012.
[38]	Z. Q. Wang, S. P. Li, Barycentric Interpolation Collocation Method for Nonlinear Problems, National Defense Industry Press, 2015.
[39]	A. M. Wazwaz, Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations, Phys. Lett. A, 352 (2006), 500–504. https://doi.org/10.1016/j.physleta.2005.12.036 doi: 10.1016/j.physleta.2005.12.036
[40]	P. Veeresha, D. G. Prakasha, Novel approach for modified forms of Camassa-Holm and Degasperis-Procesi equations using fractional operator, Commun. Theor. Phys., 72 (2020), 105002. https://doi.org/10.1088/1572-9494/aba24b doi: 10.1088/1572-9494/aba24b
[41]	J. P. Berrut, M. S. Floater, G. Klein, Convergence rates of derivatives of a family of barycentric rational interpolants, Appl. Numer. Math., 61 (2011), 989–1000. https://doi.org/10.1016/j.apnum.2011.05.001 doi: 10.1016/j.apnum.2011.05.001
[42]	X. Shen, X. H. Yang, H. X. Zhang, The high-order ADI difference method and extrapolation method for solving the two-dimensional nonlinear parabolic evolution equations, Mathematics, 12 (2024), 3469. https://doi.org/10.3390/math12223469 doi: 10.3390/math12223469

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